* European stocks down, euro near lows after inflation data
* Data seen cementing case for ECB to ease policy
* Asian shares rise as U.S., China activity picks up (Updates after inflation data)
LONDON, June 3 (Reuters) - European shares dipped on Tuesday and the euro held near 3 1/2-month lows against the dollar after a fall in euro zone inflation cemented the case for the European Central Bank to ease monetary policy later this week.
The pan-European FTSE Eurofirst 300 equity index was down 0.3 percent at 0930 GMT, barely reacting to the data that showed annual consumer price inflation unexpectedly slowed to 0.5 percent in May.
The ECB, which targets inflation of close to 2 percent, meets on Thursday and is widely expected to cut interest rates, including lowering the rate that banks are charged for depositing funds with the central bank to below zero.
The euro briefly rose after the numbers, reflecting relief that price growth had not decelerated even further, while German government bond futures fell.
"Today's inflation numbers underscore the need for the ECB to act. The ECB has consistently underestimated the deflationary forces threatening Europe and now is the time for unconventional monetary policy," said Dominic Rossi, Global Chief Investment Officer at Fidelity Worldwide Investment.
While recent data pointing to a weaker-than-expected economic recovery have weighed on stocks, the prospect of ECB intervention has offered some support.
However, expectations of lower euro zone rates, and recent upbeat U.S. economic data, have combined to push the euro to its weakest against the dollar since mid-February.
The single currency was steady at $1.3605, all but flat on the day and not far from a low of $1.3586 hit last week.
The dollar index, which measures the greenback against a basket of currencies, edged down but was close to Monday's four-month high.
The dollar stood at 102.40 yen, having earlier hit 102.49, its strongest in more than a month.
German 10-year government bond yields, which hit 12-month lows last week, rose 3.1 basis points to 1.34 percent. Bund futures declined. Some traders said the weak inflation data was already priced into the market and prompted investors to book profits after a recent rally.
"That number was if anything bond-friendly. We expect the Bund to regain its momentum and start rallying again ahead of Thursday's ECB meeting," one trader said.
U.S. 10-year Treasury yields rose on Monday to 2.54 percent after the Institute of Supply Management showed U.S. manufacturing activity accelerated in May. The ISM data helped push U.S. stocks higher, with the Dow Jones average and S&P 500 index closing at record highs.
The U.S. numbers and Chinese data showing factory and service-sector performance had their best showings in months in May helped push Asian shares higher on Tuesday.
Japan's Nikkei hit a two-month high, further boosted by talk of public pension funds increasing their assets allocated to domestic shares. MSCI's broadest index of Asia-Pacific shares outside Japan rose 0.4 percent, nearing a one-year high hit last week.
Emerging markets stayed broadly rangebound. Emerging dollar bond spreads versus Treasuries stood at 288 basis points , their tightest in 15 months. Emerging stocks rose 0.4 percent, just off recent 6-1/2 month highs
Higher U.S. and Asian shares helped steady gold after a five-day losing streak, though the metal was still near a four-month low at $1,246 an ounce.
Brent crude slipped towards $108 a barrel, reflecting weak European demand, although the Chinese data kept a floor under prices. (Additiional reporting by Jamie McGeever, Sujata Rao and Sudip Kar-Gupta in London, Hideyuki Sano in Tokyo; Editing by Mark Heinrich)